Question: Compute the ordered pair of positive integers $(x,y)$ such that

\begin{align*}
x^y+1&=y^x,\\
2x^y&=y^x+7.
\end{align*}
Solution: We substitute $a=x^y$ and $b=y^x$ to form the equations

\begin{align*}
a+1&=b,\\
2a &=b+7.
\end{align*}

Subtracting the first equation from the second, we obtain $a-1=7$, so $a=8$.

Substituting this into the first equation, we find $b=9$.

We see from $x^y=8$ and $y^x=9$ that the solution is $(x,y)=\boxed{(2,3)}$.